N-dimensional
From left to right, the square, the cube, and the tesseract. The square isbounded by 1-dimensional lines, the cube by 2-dimensional areas, and thetesseract by 3-dimensional volumes. A projection of the cube is given since itis viewed on a two-dimensional screen. The same applies to the tesseract,which additionally can only be shown as a projection even in three-dimensional space. A diagram showing the first four spatial dimensions. 1-D: Two points A and Bcan be connected to a line, giving a new line segment AB. 2-D: Two parallelline segments AB and CD can be connected to become a square, with thecorners marked as ABCD. 3-D: Two parallel squares ABCD and EFGH can beconnected to become a cube, with the corners marked as ABCDEFGH. 4-D:'''Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to becomea hypercube, with the corners marked as ABCDEFGHIJKLMNOP. In physics and mathematics, the '''dimension of a space or object is informallydefined as the minimum number of coordinates needed to specify any point withinit.12 Thus a line has a dimension of one because only one coordinate is neededto specify a point on it (for example, the point at 5 on a number line). A surfacesuch as a plane or the surface of a cylinder or sphere has a dimension of twobecause two coordinates are needed to specify a point on it (for example, to locatea point on the surface of a sphere you need both its latitude and its longitude).The inside of a cube, a cylinder or a sphere is three-dimensional because threecoordinates are needed to locate a point within these spaces. In physical terms, dimension refers to the constituent structure of all space (cf.volume) and its position in time (perceived as a scalar dimension along the t''-axis),as well as the spatial constitution of objects within—structures that correlate withboth particle and field conceptions, interact according to relative properties of mass—and are fundamentally mathematical in description. These, or other axes, may bereferenced to uniquely identify a point or structure in its attitude and relationship toother objects and occurrences. Physical theories that incorporate time, such asgeneral relativity, are said to work in 4-dimensional "spacetime", (defined as aMinkowski space). Modern theories tend to be "higher-dimensional" includingquantum field and string theories. The state-space of quantum mechanics is aninfinite-dimensional function space. The concept of dimension is not restricted to physical objects. High-dimensionalspaces occur in mathematics and the sciences for many reasons, frequently asconfiguration spaces such as in Lagrangian or Hamiltonian mechanics; theseare abstract spaces, independent of the physical space we live in. In mathematics In mathematics, the dimension of an object is an intrinsic property independent ofthe space in which the object is embedded. For example, a point on the unit circlein the plane can be specified by two Cartesian coordinates, but one can make dowith a single coordinate (the polar coordinate angle), so the circle is 1-dimensionaleven though it exists in the 2-dimensional plane. This ''intrinsic notion of dimension isone of the chief ways the mathematical notion of dimension differs from its commonusages. The dimension of Euclidean n''-space '''En'' is n. When trying to generalize to othertypes of spaces, one is faced with the question "what makes '''En'' ''n-dimensional?"One answer is that to cover a fixed ball in E''n'' by small balls of radius ε'', one needson the order of ''ε-''n'' such small balls. This observation leads to the definition of theMinkowski dimension and its more sophisticated variant, the Hausdorff dimension,but there are also other answers to that question. For example, the boundary of aball in E''n'' looks locally like E''n''-1 and this leads to the notion of the inductive dimension. While these notions agree on E''n'' , they turn out to be different whenone looks at more general spaces. A tesseract is an example of a four-dimensional object. Whereas outside ofmathematics the use of the term "dimension" is as in: "A tesseract has fourdimensions", mathematicians usually express this as: "The tesseract has dimension4", or: "The dimension of the tesseract is 4". Although the notion of higher dimensions goes back to René Descartes,substantial development of a higher-dimensional geometry only began in the 19thcentury, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfliand Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schlafi's 1852''Theorie der vielfachen Kontinuität'', Hamilton's 1843 discovery of the quaternionsand the construction of the Cayley Algebra marked the beginning of higher-dimensional geometry. The rest of this section examines some of the more important mathematicaldefinitions of the dimensions. Dimension of a vector space The dimension of a vector space is the number of vectors in any basis for thespace, i.e. the number of coordinates necessary to specify any vector. This notionof dimension (the cardinality of a basis) is often referred to as the Hameldimension or algebraic dimension to distinguish it from other notions of dimension. Manifolds A connected topological manifold is locally homeomorphic to Euclidean n''-space,and the number ''n is called the manifold's dimension. One can show that this yieldsa uniquely defined dimension for every connected topological manifold. For connected differentiable manifolds the dimension is also the dimension of thetangent vector space at any point The theory of manifolds, in the field of geometric topology, is characterized by theway dimensions 1 and 2 are relatively elementary, the high-dimensional cases n'' >4 are simplified by having extra space in which to "work"; and the cases ''n = 3 and 4are in some senses the most difficult. This state of affairs was highly marked in thevarious cases of the Poincaré conjecture, where four different proof methods areapplied. Varieties The dimension of an algebraic variety may be defined in various equivalent ways.The most intuitive way is probably the dimension of the tangent space at anyregular point. Another intuitive way is to define the dimension as the number ofhyperplanes that are needed in order to have an intersection with the variety that isreduced to a finite number of points (dimension zero). This definition is based onthe fact that the intersection of a variety with a hyperplane reduces the dimensionby one unless if the hyperplane contains the variety. An algebraic set being a finite union of algebraic varieties, its dimension is themaximum of the dimensions of its components. It is equal to the maximal length ofthe chains of sub-varieties of the given algebraic set(the length of such a chain is the number of ""). Krull dimension The Krull dimension of a commutative ring is the maximal length of chains ofprime ideals in it, a chain of length n'' being a sequence of prime ideals related by inclusion. It is stronglyrelated to the dimension of an algebraic variety, because of the naturalcorrespondence between sub-varieties and prime ideals of the ring of thepolynomials on the variety. For an algebra over a field, the dimension as vector space is finite if and only if itsKrull dimension is 0. Lebesgue covering dimension For any normal topological space ''X, the Lebesgue covering dimension of X'' isdefined to be n if ''n is the smallest integer for which the following holds: any opencover has an open refinement (a second open cover where each element is asubset of an element in the first cover) such that no point is included in more than n''+ 1 elements. In this case dim ''X = n''. For ''X a manifold, this coincides with thedimension mentioned above. If no such integer n'' exists, then the dimension of ''X issaid to be infinite, and one writes dim X'' = ∞. Moreover, ''X has dimension −1, i.e. dim''X'' = −1 if and only if X'' is empty. This definition of covering dimension can beextended from the class of normal spaces to all Tychonoff spaces merely byreplacing the term "open" in the definition by the term "'functionally open'". Inductive dimension An inductive definition of dimension can be created as follows. Consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. By dragginga 0-dimensional object in some direction, one obtains a 1-dimensional object. Bydragging a 1-dimensional object in a ''new direction, one obtains a 2-dimensionalobject. In general one obtains an (n'' + 1)-dimensional object by dragging an ''n''dimensional object in a ''new direction. The inductive dimension of a topological space may refer to the small inductivedimension or the large inductive dimension, and is based on the analogy that (n'' +1)-dimensional balls have ''n dimensional boundaries, permitting an inductivedefinition based on the dimension of the boundaries of open sets. Hausdorff dimension For structurally complicated sets, especially fractals, the Hausdorff dimension isuseful. The Hausdorff dimension is defined for all metric spaces and, unlike theHamel dimension, can also attain non-integer real values.3 The box dimension orMinkowski dimension is a variant of the same idea. In general, there exist moredefinitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values. Fractals have been found useful to describe manynatural objects and phenomena.45 Hilbert spaces Every Hilbert space admits an orthonormal basis, and any two such bases for aparticular space have the same cardinality. This cardinality is called the dimensionof the Hilbert space. This dimension is finite if and only if the space's Hameldimension is finite, and in this case the above dimensions coincide. In physics Spatial dimensions Classical physics theories describe three physical dimensions: from a particularpoint in space, the basic directions in which we can move are up/down, left/right,and forward/backward. Movement in any other direction can be expressed in termsof just these three. Moving down is the same as moving up a negative distance.Moving diagonally upward and forward is just as the name of the direction implies;i.e., moving in a linear combination of up and forward. In its simplest form: a linedescribes one dimension, a plane describes two dimensions, and a cube describesthree dimensions. (See Space and Cartesian coordinate system.) |- |2 | |- |3 | |} Time A temporal dimension is a dimension of time. Time is often referred to as the"fourth dimension" for this reason, but that is not to imply that it is a spatialdimension. A temporal dimension is one way to measure physical change. It isperceived differently from the three spatial dimensions in that there is only one of it,and that we cannot move freely in time but subjectively move in one direction. The equations used in physics to model reality do not treat time in the same waythat humans commonly perceive it. The equations of classical mechanics aresymmetric with respect to time, and equations of quantum mechanics aretypically symmetric if both time and other quantities (such as charge and parity)are reversed. In these models, the perception of time flowing in one direction is anartifact of the laws of thermodynamics (we perceive time as flowing in thedirection of increasing entropy). The best-known treatment of time as a dimension is Poincaré and Einstein'sspecial relativity (and extended to general relativity), which treats perceivedspace and time as components of a four-dimensional manifold, known asspacetime, and in the special, flat case as Minkowski space. Additional dimensions In physics, three dimensions of space and one of time is the accepted norm.However, there are theories that attempt to unify the fundamental forces byintroducing more dimensions. Superstring theory, M-theory and Bosonic string theory posit that physical space has 10, 11 and 26 dimensions, respectively. Theseextra dimensions are said to be spatial. However, we perceive only three spatialdimensions and, to date, no experimental or observational evidence is available toconfirm the existence of these extra dimensions. A possible explanation that hasbeen suggested is that space acts as if it were "curled up" in the extra dimensionson a subatomic scale, possibly at the quark/string level of scale or below. An analysis of results from the Large Hadron Collider in December 2010 severelyconstrains theories with large extra dimensions.6 Other physical theories that have introduced extra dimensions of space are: * Kaluza–Klein theory introduces extra dimensions to explain the fundamentalforces other than gravity (originally only electromagnetism). * Large extra dimension and the Randall–Sundrum model attempt to explain theweakness of gravity. This is also a feature of brane cosmology. * Universal extra dimension Networks and dimension Some complex networks are characterized by fractal dimensions.7 The concept ofdimension can be generalized to include networks embedded in space.8 Thedimension characterize their spatial constraints. Literature Perhaps the most basic way the word dimension is used in literature is as ahyperbolic synonym for feature, attribute, aspect, or magnitude. Frequently thehyperbole is quite literal as in he's so 2-dimensional, meaning that one can see at aglance what he is. This contrasts with 3-dimensional objects, which have an interiorthat is hidden from view, and a back that can only be seen with further examination. Science fiction texts often mention the concept of dimension, when really referringto parallel universes, alternate universes, or other planes of existence. This usageis derived from the idea that to travel to parallel/alternate universes/planes ofexistence one must travel in a direction/dimension besides the standard ones. Ineffect, the other universes/planes are just a small distance away from our own, butthe distance is in a fourth (or higher) spatial (or non-spatial) dimension, not thestandard ones. One of the most heralded science fiction novellas regarding true geometricdimensionality, and often recommended as a starting point for those just starting toinvestigate such matters, is the 1884 novel Flatland by Edwin A. Abbott. IsaacAsimov, in his foreword to the Signet Classics 1984 edition, described Flatland as"The best introduction one can find into the manner of perceiving dimensions." The idea of other dimensions was incorporated into many early science fictionstories, appearing prominently, for example, in Miles J. Breuer's The Appendix andthe Spectacles (1928) and Murray Leinster's The Fifth-Dimension Catapult (1931);and appeared irregularly in science fiction by the 1940s. Classic stories involvingother dimensions include Robert A. Heinlein's —And He Built a Crooked House(1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract, and Alan E. Nourse's Tiger by the Tail and''The Universe Between'' (both 1951). Another reference is Madeleine L'Engle'snovel A Wrinkle In Time (1962), which uses the 5th dimension as a way for"tesseracting the universe" or in a better sense, "folding" space to move across itquickly. The fourth and fifth dimensions were also a key component of the book The Boy Who Reversed Himself, by William Sleator. Philosophy In 1783, Kant wrote: "That everywhere space (which is not itself the boundary ofanother space) has three dimensions and that space in general cannot have moredimensions is based on the proposition that not more than three lines can intersectat right angles in one point. This proposition cannot at all be shown from concepts,but rests immediately on intuition and indeed on pure intuition a priori because it isapodictically (demonstrably) certain."9 Space has Four Dimensions, is a short story published in 1846 by Germanphilosopher and experimental psychologist Gustav Fechner (under thepseudonym Dr. Mises). The protagonist in the tale is a shadow who is aware of, andable to communicate with, other shadows; but is trapped on a two-dimensionalsurface. According to Fechner, the shadow-man would conceive of the thirddimension as being one of time.10 The story bears a strong similarity to the"Allegory of the Cave", presented in Plato's The Republic written around 380 B.C. Simon Newcomb wrote an article for the Bulletin of the American MathematicalSociety in 1898 entitled "The Philosophy of Hyperspace".11 Linda DalrympleHenderson coined the term Hyperspace philosophy in her 1983 thesis about thefourth dimension in early-twentieth-century art. It is used to describe those writersthat use higher-dimensions for metaphysical and philosophical exploration.12Charles Howard Hinton (who was the first to use the word "tesseract" in 1888)and Russian esotericist P. D. Ouspensky are examples of "hyperspacephilosophers". More dimensions * Degrees of freedom (mechanics) * Degrees of freedom (physics and chemistry) * Degrees of freedom (statistics) * Dimension of an algebraic variety * Exterior dimension * Hurst exponent * Isoperimetric dimension * Kaplan–Yorke dimension * Lebesgue covering dimension * Lyapunov dimension * Metric dimension * Pointwise dimension * Poset dimension * q-dimension; especially: ** Information dimension (corresponding to q = 1) ** Correlation dimension (corresponding to q = 2) * Vector space dimension / Hamel dimension See also * Dimension (data warehouse) and dimension tables * Dimensional analysis * Fractal dimension * Hyperspace (disambiguation page) * Intrinsic dimension * Space-filling curve A list of topics indexed by dimension * Zero dimensions: ** Point ** Zero-dimensional space ** Integer * One dimension: ** Line ** Graph (combinatorics) ** Real number * Two dimensions: ** Complex number ** Cartesian coordinate system ** List of uniform tilings ** Surface * Three dimensions ** Platonic solid ** Stereoscopy (3-D imaging) ** Euler angles ** 3-manifold ** Knot (mathematics) * Four dimensions: ** Spacetime ** Fourth spatial dimension ** Convex regular 4-polytope ** Quaternion ** 4-manifold ** Fourth dimension in art ** Fourth dimension in literature * High-dimensional topics from mathematics: ** Octonion ** Vector space ** Manifold ** Calabi–Yau spaces ** Curse of dimensionality * High-dimensional topics from physics: ** Kaluza–Klein theory ** String theory ** M-theory * Infinitely many dimensions: ** Hilbert space ** Function space References # ^''' Curious About Astronomy # '''^ MathWorld: Dimension # ^''' Fractal Dimension, Boston University Department of Mathematics andStatistics # '''^ S. Havlin, A. Bunde (1991). Fractals and Disordered Systems. Springer. # ^''' S. Havlin, A. Bunde (1994). Fractals in Science. Springer. # '''^ CMS Collaoration, "Search for Microscopic Black Hole Signatures at theLarge Hadron Collider," # ^''' C.M. Song, S. Havlin, H.A. Makse; Havlin, Shlomo; Makse, Hern�n A.(2005). "Self-similarity of complex networks". Nature '''433 (7024): 7024.arXiv:cond-mat/0503078v1. Bibcode:2005Natur.433..392S.doi:10.1038/nature03248. # ^''' D. Li, K. Kosmidis, A. Bunde, S. Havlin; Kosmidis, Kosmas; Bunde, Armin;Havlin, Shlomo (2011). "Dimension of spatially embedded networks".Nature Physics '''7 (6): 481. Bibcode:2011NatPh...7..481D.doi:10.1038/nphys1932. # ^''' Prolegomena, § 12 # '''^ Banchoff, Thomas F. (1990). "From Flatland to Hypergraphics: Interacting with Higher Dimensions". Interdisciplinary Science Reviews 15(4): 364. doi:10.1179/030801890789797239. # ^''' Newcomb, Simon (1898). "The Philosophy of Hyperspace". Bulletin ofthe American Mathematical Society '''4 (5): 187. doi:10.1090/S0002-9904-1898-00478-0. # ^ Kruger, Runette (2007). "Art in the Fourth Dimension: Giving Form to Form – The Abstract Paintings of Piet Mondrian". Spaces of Utopia: anElectronic Journal (5): 11. Further reading * Edwin A. Abbott, (1884) Flatland: A Romance of Many Dimensions, PublicDomain. Online version with ASCII approximation of illustrations at Project Gutenberg. * Thomas Banchoff, (1996) Beyond the Third Dimension: Geometry, ComputerGraphics, and Higher Dimensions, Second Edition, Freeman. * Clifford A. Pickover, (1999) Surfing through Hyperspace: UnderstandingHigher Universes in Six Easy Lessons, Oxford University Press. * Rudy Rucker, (1984) The Fourth Dimension, Houghton-Mifflin. * Michio Kaku, (1994) Hyperspace, a Scientific Odyssey Through the 10th Dimension, Oxford University Press.Category:Calculus Category:Crash Courses Category:Algebra